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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdmsscn | Structured version Visualization version GIF version |
Description: π is a subset of β. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
dvdmsscn.s | β’ (π β π β {β, β}) |
dvdmsscn.x | β’ (π β π β ((TopOpenββfld) βΎt π)) |
Ref | Expression |
---|---|
dvdmsscn | β’ (π β π β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restsspw 17321 | . . . 4 β’ ((TopOpenββfld) βΎt π) β π« π | |
2 | dvdmsscn.x | . . . 4 β’ (π β π β ((TopOpenββfld) βΎt π)) | |
3 | 1, 2 | sselid 3946 | . . 3 β’ (π β π β π« π) |
4 | elpwi 4571 | . . 3 β’ (π β π« π β π β π) | |
5 | 3, 4 | syl 17 | . 2 β’ (π β π β π) |
6 | dvdmsscn.s | . . 3 β’ (π β π β {β, β}) | |
7 | recnprss 25291 | . . 3 β’ (π β {β, β} β π β β) | |
8 | 6, 7 | syl 17 | . 2 β’ (π β π β β) |
9 | 5, 8 | sstrd 3958 | 1 β’ (π β π β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 β wss 3914 π« cpw 4564 {cpr 4592 βcfv 6500 (class class class)co 7361 βcc 11057 βcr 11058 βΎt crest 17310 TopOpenctopn 17311 βfldccnfld 20819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 ax-resscn 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-rest 17312 |
This theorem is referenced by: dvxpaek 44271 etransclem17 44582 etransclem18 44583 etransclem20 44585 etransclem21 44586 etransclem22 44587 etransclem29 44594 etransclem31 44596 etransclem34 44599 etransclem43 44608 etransclem46 44611 |
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